## Pulley & Belt

In the figure shown, a pulley is driven by a belt. What minimum coefficient of friction will prevent slipping between pulley and belt? Ignore centrifugal effects. Hint
Belt Friction:
$$F_1=F_2e^{\mu \theta}$$$where $$F_1$$ is the applied force in the direction of impending motion, $$F_2$$ is the applied force resisting impending motion, $$\mu$$ is the static coefficient of friction, and $$\theta$$ is the total angle of contact between surfaces in radians. Hint 2 Unit Circle: Belt Friction: $$F_1=F_2e^{\mu \theta}$$$
where $$F_1$$ is the applied force in the direction of impending motion, $$F_2$$ is the applied force resisting impending motion, $$\mu$$ is the static coefficient of friction, and $$\theta$$ is the total angle of contact between surfaces in radians. Recall a unit circle to determine $$\theta$$ : Therefore,
$$1,000N=(550N)e^{\mu (1.5\pi-0.5\pi)}$$$Solving for the coefficient of friction: $$ln(\frac{1,000}{550})=ln(e^{\mu \pi})=\mu \pi$$$
$$\mu=\frac{1}{\pi}\cdot ln(\frac{1,000}{550})=\frac{1}{\pi} \cdot 0.5978=0.19$$\$
0.19